Juraj Kolnik and Nathan Horton also scored for Florida, which has won seven of its last eight home games against the Lightning. Olli Jokinen had two assists for the Panthers.
Vincent Lecavalier and Ryan Clark scored for Tampa Bay and Johan Holmqvist had 24 saves.
Bouwmeester's power-play goal gave Florida a 3-2 lead. His shot from just inside the blue-line above the right circle slid past Holmqvist, who was screened on the play.
Clark tied the game 2-2 in the second period when he scored while on his side. Nikita Alexeev passed to Clark in front, who fell on his left side but was able to sweep the loose puck into the net.
Horton gave the Panthers a 2-1 lead when he took a pass from Jokinen in front and poked the puck past a diving Holmqvist at 19:11 of the first.
Kolnik tied the score 1-1 when he one-time a pass from Martin Gelinas at 14:34.
Tampa Bay scored on its first shot on goal when Lecavalier beat Auld high on the glove side from the top of the right circle.
Notes: Florida C Joe Nieuwendyk missed his second straight game due to a strained hip flexor. He plans to return to the lineup on Wednesday at Washington. ... The Lightning have scored just seven goals through their first four games. ... The Lightning are only 1-for-27 on the power play, including 0-for-20 over the last three games.